Riley has a farm on a rectangular piece of land that is $200$ meters wide. This area is divided into two parts: A square area where she grows avocados (whose side is the same as the length of the farm), and the remaining area where she lives. Every week, Riley spends $\$3$ per square meter on the area where she lives, and earns $\$7$ per square meter from the area where she grows avocados. That way, she manages to save some money every week. Write an inequality that models the situation. Use $l$ to represent the length of Riley's farm.
Solution: The strategy We know that Riley saves some money each week. In other words, her weekly earnings are more than her weekly spendings. So, if $X$ is the amount of money she earns from her avocados each week and $Y$ is the amount of money she spends on her home each week, then $X>Y$. Now, let's express $X$ and $Y$ in terms of $l$. Expressing the amount of money earned Since the region in which Riley grows avocados is a square, and since $l$ is the length of one side of this region, its area is $l^2$ square meters. Each week, Riley earns $\$7$ per square meter on this region, and so Riley earns $7l^2$ dollars per week. Expressing the amount of money spent Since Riley's farm is $200$ meters wide, and since the avocado farm uses $l$ meters of that width, the residential region of her farm is $200-l$ meters wide. Its length is of course $l$ meters, and so the total area of the residential region is $l(200-l)$ square meters. Each week, Riley spends $\$3$ per square meter on the area in which she lives, and so Riley spends $3l(200-l)$ dollars per week. Putting things together We found that $X=7l^2$ and $Y=3l(200-l)$. Since $X>Y$, we can substitute and find an inequality in terms of $l$ that models the situation. The answer is: $ 7l^2>3l(200-l)$